Triangle Sum Properties & Properties of Isosceles Triangles

-Classify triangles and find measures of their angles.- Discover the properties of

Isosceles Triangles.

Classification By Sides

Classification By Angles

Classifying Triangles

In classifying triangles, be as specific as possible.

Acute,Scalene

Obtuse,Isosceles

Triangle Sum Theorem **NEW

The sum of the measures of the interior angles of a triangle is 180o.

32

1

m<1 + m<2 + m<3 = 180°

The sum of all the angles equals 180º degrees.

90º 30º

60º

60º90º30º+

180º

Property of triangles

60º60º60º+

180º60º 60º

60º

The sum of all the angles equals 180º degrees.

Property of triangles

What is the missing angle?

70º70º

?+

180º70º 70º

?

180 – 140 = 40˚

90º30º

?+

180º30º 90º

?

180 – 120 = 60˚

What is the missing angle?

60º60º

?+

180º60º 60º

?

189 – 120 = 60˚

What is the missing angle?

30º78º

?+

180º78º 30º

?

180 – 108 = 72˚

What is the missing angle?

45x 10x

35x

90°, 70°, 20°

Find all the angle measures

180 = 35x + 45x + 10x

180 = 90x

2 = x

What can we find out?

The ladder is leaning on the ground at a 75º angle. At what angle is the top of the ladder touching the building?

75

180 = 75 + 90 + x

180 = 165 + x15˚ = x

Corollary to Triangle Sum Theorem

A corollary is a statement that readily follows from a theorem.

The acute angles of a right triangle are complementary.

m A + m B = 90∠ ∠ o

The tiled staircase shown below forms a right triangle.

The measure of one acute angle in the triangle is twice the measure of the other angle.

Find the measure of each acute angle.

Find the missing angles.

Con’t

Find the missing angles.

2x + x = 90

3x = 90

x = 30˚

2x = 60˚

SOLUTION:

Find the missing angles.

2x + (x – 6) = 90˚

3x – 6 = 90

3x = 96

x = 32

2x = 2(32) = 64˚

(x – 6) = 32 – 6 = 26˚

Isosceles Triangle at least two sides have the same length

5 m

9 in9 in

4 in

5 m

5 m

3 miles 3 miles

4 miles

Properties of an Isosceles Triangle

Has at least 2 equal sides

Has 2 equal angles

Has 1 line of symmetry

Parts of an Isosceles Triangle:

The vertex angle is the

angle between two congruent

sides

The base angles are the angles opposite the

congruent sides

Parts of an Isosceles Triangle:

The base is the side

opposite the vertex angle

Parts of an Isosceles Triangle:

Isosceles Triangle Conjecture If a triangle is isosceles, then base angles

are congruent.

If then

Converse of Isosceles Triangle Conjecture If a triangle has two congruent angles,

then it is an isosceles triangle.

If then

Equilateral Triangle Triangle with all three sides are

congruent

7 ft 7 ft

7 ft

Equilateral Triangle Conjecture An equilateral triangle is equiangular, and

an equiangular triangle is equilateral.

Find the missing angle measures.

mb =

68˚

44˚ 68˚ a

b<68° and < a are base angles

they are congruent

ma =

m<b = 180 – 68 - 68

m<b = 180 -136

Triangle sum to find <b

 

mc =

md =

Find the missing angle measures.

30.5˚

30.5˚

119˚

c d

Triangle sum = 180°180 = 119 + c + d180 – 119 = c + d61 = c + d

<c & <d are base angles and are congruent

<c = ½ (61) = 30.5<d = ½ (61) = 30.5

 

mE =

mF =

mG =

Find the missing angle measures.

60˚

60˚

60˚ GF

EEFG is an equilateral triangle<E = <F = <G

180 /3 = 60

 

Find mG.

Thus m<G = 22 + 44 = 66°And m<J = 3(22) = 66°

x = 22

Find the missing angle measures.

∆GHJ is isosceles< G = < J

x + 44 = 3x44 = 2x

 

Find mN

Thus m<N = 6(8) = 48°.m<P = 8(8) – 16 = 48°

Find the missing angle measures.

6y = 8y – 16-2y = -16

y= 8

Base angles are =

Using Properties of Equilateral Triangles

Find the value of x.

∆LKM is equilateral m<K = m<L = m<M

Find the missing angle measures.

180/3 = 60°

2x + 32 = 602x = 37

x = 18.5°

 

Using Properties of Equilateral Triangles

Find the value of y.

∆NPO is equiangular∆NPO is also equilateral.

Find the missing side measures.

5y – 6 = 4y +12y – 6 = 12

y = 18

Side NO = 5(18) – 6 = 90ft

ftft

 

Using the symbols describing shapes answer the following questions:

36o a

b

c

45o

d

Isosceles triangleTwo angles are equal

a = 36o

b = 180 – (2 × 36) = 108o

Equilateral triangleall angles are equal

c = 180 ÷ 3 = 60o

Right-angled triangle

d = 180 – (45 + 90) = 45o

Find the missing angle measures.

a = 64o

b = 180 – (2 ×64o ) = 52o

c = dc + d = 180 - 72

c + d = 108

c = d = 54o

Equilateral trianglee = f = g = 60o

h = ih + i = 180 - 90h + i = 90h = i = 45o

Find the missing angle measures.

A

A

B C D

B C

D

p = 50o

q = 180 – (2 ×50o ) = 80o

r = q = 80o vertical angles are equal

Therefore : s = t = p = 50o

Properties of Triangles

a = b= c = 60o

d = 180 – 60 = 120o

e + 18 = a = 60

exterior angle = sum of remote interior angles

e = 60 – 18 = 42o

p = q = r = 60o

s = t = 180 - 43 = 68.5o

2

Find the missing angle measures.

1) Find the value of x

2) Find the value of y

Find the missing angle measures.

1) x is a base angle180 = x + x + 50

130 = 2xx = 65°

2) y & z are remote interior angles and base angles of an isosceles

triangleTherefore: y + z = x and y = z

y + z = 80°y = 40°

50°

x°y°

D

C

B

A

z

1) Find the value of x

2) Find the value of y

50E

DB

CA

yx

Find the missing angle measures.

2) y is the vertex angle

y = 180 – 100y = 80°

1) ∆CDE is equilateralAll angles = 60°

Using Linear Pair <BCD = 70°

x is the vertex anglex = 180 – 70 – 70

x = 40°

60°70°

HomeworkIn your textbook:

Lesson 4.1/ 1-9; 4.2/ 1-10